Semilinearity and Context-Freeness of Languages Accepted by Valence Automata

Valence automata are a generalization of various models of automata with storage. Here, each edge carries, in addition to an input word, an element of a monoid. A computation is considered valid if multiplying the monoid elements on the visited edges yields the identity element. By choosing suitable monoids, a variety of automata models can be obtained as special valence automata. This work is concerned with the accepting power of valence automata. Specifically, we ask for which monoids valence automata can accept only context-free languages or only languages with semilinear Parikh image, respectively. First, we present a characterization of those graph products (of monoids) for which valence automata accept only context-free languages. Second, we provide a necessary and sufficient condition for a graph product of copies of the bicyclic monoid and the integers to yield only languages with semilinear Parikh image when used as a storage mechanism in valence automata. Third, we show that all languages accepted by valence automata over torsion groups have a semilinear Parikh image.